Multi-dimensional Simulation of Phase Change by a 0D-2D Model Coupling via Stefan Condition

doi:10.1007/s42967-021-00157-y

Abstract

Abstract: Considering phase changes associated with a high-temperature molten material cooled down from the outside, this work presents an improvement of the modelling and the numerical simulation of such processes for an application pertaining to the safety of light water nuclear reactors. Postulating a core meltdown accident, the behaviour of the core melt (aka corium) into a steel vessel is of tremendous importance when evaluating the vessel integrity. Evaluating correctly the heat fluxes requires the numerical simulation of the interaction between the liquid material and its solid counterpart which forms during the solidification process, but also may melt back. To simulate this configuration, encountered in various industrial applications, one considers a bi-phase model constituted by a liquid phase in contact and interaction with its solid phase. The liquid phase may solidify in presence of low energetic source, while the solid phase may melt due to an intense heat flux from the high-temperature liquid. In the frame of the in-house legacy code, several simplifying assumptions (0D multi-layer discretization, instantaneous heat transfer via a quadratic temperature profile in solids) are made for the modelling of such phase changes. In the present work, these shortcomings are illustrated and further overcome by solving a 2D heat conduction model in the solid by a mixed Raviart-Thomas finite element method coupled to the liquid phase due to heat and mass exchanges through Stefan condition. The liquid phase is modeled with a 0D multi-layer approach. The 0D-liquid and 2D-solid models are coupled by a Stefan like phase change interface model. Several sanity checks are performed to assess the validity of the approach on 1D and 2D academical configurations for which exact or reference solutions are available. Then more advanced situations (genuine multi-dimensional phase changes and an "industrial-like scenario") are simulated to verify the appropriate behavior of the obtained coupled simulation scheme.

Key words: Simulation of phase change, Fusion, Solidification, 0D multi-layer model, 2D heat conduction model, Model coupling

CLC Number:

Adrien Drouillet, Romain Le Tellier, Raphaël Loubère, Mathieu Peybernes, Louis Viot. Multi-dimensional Simulation of Phase Change by a 0D-2D Model Coupling via Stefan Condition[J]. Communications on Applied Mathematics and Computation, 2023, 5(2): 853-884.

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References

1. Alnæs, M.S., Blechta, J., Hake, J., Johansson, A., Kahlet, B., Logg, A., Richardson, C., Ring, J., Rognes, M.E., Wells, G.N.,:The FEniCS project version 1.5. Arch. Numer. Softw. 3(100), 9-23 (2015)
2. Auricchio, F., Beirão da Veiga, L., Brezzi, F., Lovadina, C.:Mixed Finite Element Methods. American Cancer Society, Atlanta (2017)
3. Budhia, H., Kreith, F.:Heat transfer with melting or freezing in a wedge. Int. J. Heat Mass Transf. 16(1), 195-211 (1973)
4. Carénini, L., Fichot, F., Bakouta, N., Le Tellier, R., Viot, L., Melnikov, I., Pandazis, P., Filippov, A.:Main outcomes from the IVR code benchmark performed in the IVMR project. In:Proc. of the 9th European Review Meeting on Severe Accident Research ERMSAR2019, Prague, Czech Republic (2019)
5. Carénini, L., Fichot, F., Seignour, N.:Modelling issues related to molten pool behaviour in case of invessel retention strategy. Ann. Nucl. Energy 118, 363-374 (2018)
6. Chen, S., Merriman, B., Osher, S., Smereka, P.:A simple level set method for solving Stefan problems. J. Comput. Phys. 135(1), 8-29 (1997)
7. Esmali, H., Khatib-Rahbar, M.:Analysis of likelihood of lower head failure and ex-vessel fuel coolant interaction energetics for AP1000. Nucl. Eng. Des. 235, 1583-1605 (2005)
8. Fauske and Associates Inc. MAAP4-modular accident analysis program for LWR power plants:code structure and theory. Technical report, Electric Power Research Institute (1994)
9. Gabriel, A., Li, D., Chiocchetti, S., Tavelli, M., Peshkov, I., Romenski, E., Dumbser, M.:A unified first order hyperbolic model for nonlinear dynamic rupture processes in complex diffuse fracture zones. Philos. Trans. R. Soc. A 379, 20200130 (2021)
10. Gupta, S.C.:Chapter 1-the Stefan problem and its classical formulation. In:Gupta, S.C. (ed) The Classical Stefan Problem, 2nd edn, pp. 1-35. Elsevier, Amsterdam (2018)
11. Hosseinzadeh, H., Moghadam, A.J.:Thermal simulation of solidification process in continuous casting. Int. J. Eng. 28, 812-821 (2015)
12. Jacquemain, D.:Les accidents de fusion du coeur des réacteurs nucléaires de puissance:état des connaissances (French). In:Collection Sciences et Techniques, EDP Sciences, Les Ulis, France (2013)
13. Kamenomostskaya, S.L.:On Stefan problem. Mat. Sb. (in Russian) 53, 489-514 (1961)
14. Kucharik, M., Shashkov, M., Wendroff, B.:An efficient linearity-and-bound-preserving remapping method. J. Comput. Phys. 188(2), 462-471 (2003)
15. Le Tellier, R., Saas, L., Bajard, S.:Transient stratification modelling of a corium pool in a LWR vessel lower head. Nucl. Eng. Des. 287, 68-77 (2015) (ISSN 0029-5493)
16. Le Tellier, R., Skrzypek, E., Saas, L.:On the treatment of plane fusion front in lumped parameter thermal models with convection. Appl. Therm. Eng. 120, 314-326 (2017)
17. Le Tellier, R., Tiwari, V., Habert, B., Bakouta, N.:Consistent use of Calphad data for in-vessel corium pool modeling:some analytical and practical considerations. In:Proc. of the 9th European Review Meeting on Severe Accident Research ERMSAR2019, Prague, Czech Republic (2019)
18. Logg, A., Mardal, K.-A., Wells, G.N., et al.:Automated Solution of Differential Equations by the Finite Element Method. Springer, Berlin (2012)
19. Loubère, R., Shashkov, M.J.:A subcell remapping method on staggered polygonal grids for arbitrary-Lagrangian-Eulerian methods. J. Comput. Phys. 209(1), 105-138 (2005)
20. Mathieu, P., Le Tellier, R., Viot, L., Drouillet, A., Saas, L.:Hybrid lumped/distributed parameter model for treating the vessel lower head ablation by corium during a LWR severe accident. In:The 27th International Conference Nuclear Energy for New Europe, Portoroz, Slovenia (2018)
21. Rathjen, K.A., Jiji, L.M.:Heat conduction with melting or freezing in a corner. ASME. J. Heat Transf. 93(1), 101-109 (1971)
22. Raviart, P.A., Thomas, J.M.:A mixed finite element method for 2-nd order elliptic problems. In:Galligani, I., Magenes, E. (eds) Mathematical Aspects of Finite Element Methods, vol. 606, pp. 292-315. Springer, Berlin, Heidelberg (1977). https://doi.org/10.1007/BFb0064470
23. Rempe, J.L., Knudson, D.L., Allison, C.M., Thinnes, G.L., Atwood, C.L.:Potential for AP600 invessel retention through ex-vessel flooding. Technical Report:INEEL/EXT-97-00779, USA (1997)
24. Rempe, J.L., Knudson, D.L., Cebull, M., Atwood, C.L.:Potential for in-vessel retention through ex- vessel flooding. In:Proc. of the OECD/CSNI Workshop on in-Vessel Core Debris Retention and Coolability, Germany (1998)
25. Sehgals, B.R.:Nuclear Safety in Light Water Reactors:Severe Accident Phenomenology. Elsevier/Academic Press, Amsterdam (2012)
26. Shams, A., Dovizio, D., Zwijsen, K., Le Guennic, C., Saas, L., Le Tellier, R., Peybernes, M., Bigot, B., Skrzypek, E., Skrzypek, M., Vyskocil, L., Carenini, L., Fichot, F.:Status of computational fluid dynamics for in-vessel retention:challenges and achievements. Ann. Nucl. Energy 135, 107004 (2020)
27. Sidorov, V., Bezlepkin, V., Astafyeva, V., Kukhtevich, V., Semashko, S.:Numerical and experimental researches on in-vessel retention possibility for VVER reactors. In:Proc. of the 7th European Review Meeting on Severe Accident Research ERMSAR-2015, Marseille, France (2015)
28. Stefan, J.:Über die theorie der eisbildung. Mon. Mat. Phys. 1, 1-6 (1890)
29. Summers, R.M., Cole, Jr, R.K., Smith, R.C., Stuart, D.S., Thompson, S.L., Hodge, S.A., Hyman, C.R., Sanders, R.L.:MELCOR computer code manuals. Technical Report:NUREG/CR-6119-Vol. 2; SAND-93-2185-Vol. 2, USA (1995)
30. Theofanous, T.G., Liu, C., Addition, S., Angelini, S., Kymalainen, O., Salmassi, T.:In-vessel coolability and retention of a core melt. Nucl. Eng. Des. 169, 1-48 (1997)
31. Tuomisto, H., Theofanous, T.G.:A consistent approach to severe accident management. Nucl. Eng. Des. 148, 171-183 (1994)
32. Van Dorsselaere, J.-P., Seropian, C., Chatelard, P., Jacq, F., Fleurot, J., Giordano, P., Reinke, N., Schwinges, B., Allelein, H.-J., Luther, W.:The ASTEC integral code for severe accident simulation. Nucl. Technol. 165, 293-307 (2009)
33. Viot, L.:Couplage et synchronisation de modèles dans un code scénario d'accidents graves dans les réacteurs nuclésaires. PhD thesis, 2018. Thèse de doctorat dirigée par de Vuyst, Florian Mathématiques appliquées Université Paris-Saclay (ComUE) (2018)
34. Viot, L., Le Tellier, R., Peybernes, M.:Modelling of the corium crust of a stratified corium pool during severe accidents in light water reactors. Nucl. Eng. Des. 368, 110816 (2020)
35. Viot, L., Saas, L., De Vuyst, F.:Solving coupled problems of lumped parameter models in a platform for severe accidents in nuclear reactors. Int. J. Multisc. Comput. Eng. 16, 555-577 (2018)
36. Voller, V.R.:An overview of numerical methods for solving phase change problems. Adv. Numer. Heat Transf. 1(9), 341-380 (1997)
37. Wang, J., Zhang, X.:Coupled solid-liquid phase change and thermal flow simulation by particle method. Int. Commun. Heat Mass Transf. 113, 104 (2020)
38. Wu, H., Liu, Z., Yao, Y.:Numerical modelling for solid-liquid phase change phenomena in porous media:shell-and-tube type latent heat thermal energy storage. Appl. Energy 112, 1222-1232 (2013)
39. Yang, H., He, Y.:Solving heat transfer problems with phase change via smoothed effective heat capacity and element-free Galerkin methods. Int. Commun. Heat Mass Transf. 37(4), 385-392 (2010)
40. Zanella, R., Le Tellier, R., Plapp, M., Tegze, G., Henry, H.:Three-dimensional numerical simulation of droplet formation by Rayleigh-Taylor instability in multiphase corium. Nucl. Eng. Des. 379, 111- 177 (2021)

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